Will the real Investment Risk please stand up?

Introduction

How much risk are you taking with your investments? There is no shortage of answers to this question. In fact, there are too many answers. Not only does the answer depend on who you ask, the way risk is measured depends on the investment under consideration. For example, when discussing a mutual fund, the answer could be 'That's low risk. This fund has low volatility and a high Sharpe Ratio.' When it comes to real estate investments, you never hear about volatility or Sharpe Ratios or any of the other traditional ways of measuring investment risk. Surely investing in real estate involves some risk. So how is a rational person supposed to compare the risk and reward of a real estate investment to that of a mutual fund?

In this article we will discuss to what extent these traditional risk measures are an appropriate way to think about investment risk. We will explain why this approach does not work for less liquid instruments such as real estate and propose a less rigorous but much more universal way to estimate investment risk that allows us to compare very different types of investment vehicles on an equal footing.

Our investment choices tend to include illiquid instruments ranging from our primary residence to a private company we may own part of. Clearly it is important to be able to compare these investments to more liquid investments in stocks and bonds.

Illiquid Instruments

One problem with traditional risk measures is that we need to know the value of our investment at regular time intervals. This is easy to do with a stock like IBM. For example, we could capture the official closing price each day. It is much more difficult to do with illiquid instruments. There is no easy way to determine the value of your primary residence at the end of each business day. In fact the only way to really determine the value is to sell it, which is clearly not something that could be done regularly.

The table on the right illustrates the problem. Please refer to the math refresher for information on how the calculations are done. The table shows the value of a generic asset at different times. We are assuming that we know the values in red, perhaps because somebody else traded a similar asset for this price. We also assume that we have no information about the value of the assets at any other time. In order to calculate traditional risk measures we need the price of the asset at regular time intervals, for example daily. The two sides of the table show two different ways of filling in estimated prices. On the left we assume that the price increases by $1 each day and on the right we assume that it stays constant until our observations tell us that it has changed. Both are reasonable assumptions, but when we calculate the volatility and Sharpe Ratio for the two cases we get very different answers. Clearly we cannot use the Sharpe Ratio to analyze the risk-adjusted return of an illiquid investment, because the answer depends strongly on data we do not have.

This example explains why nobody talks about the Sharpe Ratio of their house, even though Sharpe Ratios of mutual funds and other liquid investments are frequently used to make investment choices. We will discuss some other issues with Sharpe Ratios before proposing a different way to look at risk-adjusted returns that allows us to compare a house to a mutual fund on an equal footing.

Non-Normal Distributions

Let us consider one of the most liquid investments next to see what can go wrong even it we do not run into the problems discussed in the previous section. The daily closing values of the S&P 500 are readily available. It should be the poster child of traditional risk measures, but in reality that is not how it turns out.

The average daily index return since 1950 is 0.03% (8.7% annually). The volatility (standard deviation) of these returns is 0.89% and the Sharpe Ratio is Sqrt(252)*0.03%/0.89% = 0.54. The S&P 500 returns 54% of a typical annual fluctuation per year and there should be only about a 2.5% chance of seeing a one-day loss of more than two standard deviation (-1.78%). The assumption that underlies these statements is that the distribution of returns is a Normal Distribution. The graph shows the distribution of daily returns since 1950. Most of it looks like the familiar bell curve. The axis extends so far left because the largest negative one-day return was nearly -20%.

Let us take a look at the largest negative one-day returns and calculate the probability of such returns occurring if the return distribution is really normal. The table show the six worst days for the S&P 500, how many standard deviations the move was, the probability of such a move occurring based on the assumption that the distribution is Normal, the number of years that should pass between events of this magnitude and finally the date of the returns.

Perhaps the most striking data in the table is the expected number of years between events of this size and the number of times such events have happened in the last 50 years. During the last 50 years we have had at least six down days of a magnitude that should happen only every 100+ billion years. Obviously the assumption that the returns of the S&P 500 are normally distributed is incorrect. There is a much higher probability of a large move than the Normal Distribution would suggest.

What does this imply for our calculation of the risk-adjusted return based on the Sharpe Ratio? In that calculation we used the standard deviation as a measure of risk in the denominator, but our analysis of historical data shows that the standard deviation significantly underestimates the true risk of investing in the S&P 500. The real risk-adjusted return is significantly lower than our original calculation indicates, but the traditional framework does not provide a recipe for calculating it.

What now?

We have seen that traditional ways of measuring risk and risk adjusted returns fail in certain situations. They do not apply very well to illiquid investments that cannot easily be valued at regular time intervals. They also tend to underestimate the magnitude and frequency of large adverse moves. Many of our investment choices are illiquid and practically all of them involve risks that are not fully captured with the standard assumptions of conventional risk measures.

Does this mean that we should discard conventional ways of thinking about risk? No! But we should supplement the traditional approach with other measures that take extreme events into account. The Sharpe Ratio and similar risk measures work quite well most of the time. Outside of rather extreme market conditions these measure provide a good framework for analyzing investments. Our examples only illustrate that by themselves they are incomplete, not that they are useless or incorrect.

Generally it will be necessary to incorporate extreme events and illiquid investments in a complete risk analysis. The framework that we propose here is a crude, but fairly effective way of doing so. An added benefit is that it works equally well for liquid and illiquid instruments. An analysis based on this framework looks at risk from a different angle. Combined with the traditional approach it provides a more complete assessment of investment risk.

Back to the Basics

The reason why we study risk is the possibility of investment losses. The real question we need to answer is 'How much can I lose if things go wrong?' Then we need to balance the answer to that question with an estimate of how much we would make if the investment works out as expected. Let us see how far we can take this simple-minded approach using the S&P 500 as an example.

Q: How much can I lose if I invest in the S&P 500?

A: At least as much as you would have lost in any of the historical market crashes. The S&P 500 lost about 20% in a single day during the crash of '87. We know that the one-day risk of an investment in the S&P 500 is at least that much, but of course there is no guarantee that the next crash won't be bigger. Nonetheless, a 20% loss is probably a good ballpark figure for a worst case scenario.

Q: How much should I expect to make if I invest in the S&P 500?

A: This is generally an easier question to answer. Historical returns for various asset classes tend to be fairly stable. Assuming that the average return going forward will be similar to historical returns should give us a good estimate. It is important to average over several market cycles to get usable estimates. The average annual return over the last 50 years was 8.7%. It is reasonable to assume a similar return for the next 50 years.

Now that we have a risk and a return estimate, it is natural to calculate a risk-adjusted return similar to the Sharpe Ratio by dividing the expected annual return by the worst case risk:

(annual return)/(worst case) = 8.7% / 20% = 0.435

We will refer to this ratio as the Tail Risk Ratio (TRR) because the risk component of it looks at extreme or tail risk. The TRR has a simple interpretation. It takes about 2.3 normal years (= 1/0.435) to recover from a worst-case crash. This way of looking at the relation between risk and reward incorporates the risk due to large events that traditional risk measures do not fully capture and, as we will see below, it works for illiquid instruments too. Unfortunately it also has some serious drawbacks that keep us from discarding the traditional risk measures in favor of this ratio.

Chasing Tail Risk

The most serious drawback of the TRR is the lack of rigor when estimating the magnitude of the worst case. For traditional risk measures there is a well defined mathematical recipe, but statistics offers no guidance when it comes to estimating the properties of very rare events. Making a reasonable estimate is more an art than a science, but there are some guidelines we can use.

First, historical data cannot tell us what the worst case looks like, but if our estimate is milder than actual historical crashes, it must be flawed.

Second, we always should make several different estimates of the worst case scenario. Since this is not a precise science, the answers are bound to differ quite a bit, but usually they cluster around some central value. If they do, we can use that value as our risk estimate and if they don't it is probably a good indication that we are missing some risk factors in at least some of the estimates.

Let us analyze the tail risk of the S&P 500 in more detail to illustrate how to construct multiple risk estimates. The following three estimate give differing results:

1. Use the worst one-day return: -20%. This is the estimate we used above.
2. Find the worst one-day return for each industry sector in the index and assume that they all occur on the same day. Lining up the "sector crashes" like this would result in a 7.8% one-day loss. This estimate is based on an equal weight for each industry and only uses data going back to 1998.
3. Assume that the two largest companies in the S&P 500 (GE and XOM) go bankrupt on the same day. This would result in about a 6% loss for the S&P 500.

On the face of it, estimates 2 and 3 are quite pessimistic. It is hard to imagine an economic situation where all sectors experience their worst one-day loss and it is highly unlikely that both General Electric and Exxon Mobil will go bankrupt on the same day. Yet the risk estimates based on these assumptions are much smaller than real historical one-day moves. It looks like we are missing a significant risk factor in estimates 2 and 3. Or is 20% too high? Was there something unique about that event that is unlikely to happen again?

Market crashes are fairly common. We do not have to rely on the '87 crash to estimate how large a one-day move can be. Some examples of other crashes include a 14% decline of the DOW in '28 and a 23% decline in '87, a 15% decline of the Nikkei in '87and a 10% decline in '53, and finally a 10% decline of the NASDAQ in '00. While 20% is on the high side of the range, one-day declines well in excess of 10% can clearly happen in different markets and at different times.

Something must be wrong with your estimates 2. and 3. We are missing an important risk factor that is characteristic of market crashes.

The graph shows what it is. The bars represent the one-day returns of the stocks that make up the DOW during the crash of '87. The best performer was down about 10% and the worst performers were down around 30%. In effect the stocks in the DOW provided no diversification! Diversification tends to disappear during market crashes because panicked sellers decide to sell all of their holdings, not just specific stocks. Our estimates 2. and 3. implicitly assume that there is at least some level of diversification and are therefore incomplete.

Estimate 2. does not correctly include diversification because the sector returns we used do not include data from a market crash. Outside of market crashes diversification exists and the returns we used reflect its presence.

Estimate 3. explicitly assumes diversification. While GE and XOM loose 100% of their value, the remaining stocks in the S&P 500 do not move.

Now that we know what was missing before, we can make another estimate that incorporates the fact that in market crashes there is no meaningful diversification. A typical bad day for a stock is a loss of about 20%, perhaps as a result of a poor earnings announcement, a lawsuit, and SEC investigation or any number of other things. This shows us what happens if investors panic in a stock specific manner. The difference between a normal day and a market crash is how far the panic spreads. If it only affects a few stocks, most people will never notice, but if it spreads to all stocks, it turns into a market crash. Since a single stock panic costs about 20% we should expect a market crash to result in about the same decline. This risk estimate is in line with historical crashes and even provides a reasonable explanation of what makes a market crash.

This example shows that making reasonable tail risk estimates requires more effort than just calculating the standard deviation of a return series. However it clarifies our understanding of the nature of the risks and it is an important step towards deciding if a given investment is worth making.

Other Examples

We have seen that the TRR provides a way to analyze the risk adjusted return of the S&P 500. The second drawback of traditional risk measures, the lack of universality, has not been addressed so far. We will remedy that by briefly discussing a few different examples. It is important to point out that the tail risk estimates in these examples are for illustration purposes only and not based on a careful analysis of historical data. The actual magnitude of these risks could be far greater.

A house as an investment (Take 1)

We pay the full price of $100,000 in cash for the house. Average historical returns for real estate are around 7% per year. This gives us the expected return. Now we need to estimate how much we can lose if the real estate market collapses. For this example we will use 10% as an estimate of the maximum decline of the real estate market over the period of time it takes to sell the house. Let us further assume that we have to sell the house 5% below market to move it quickly. This puts the maximum loss at 15% (We are not including closing costs in this estimate, which can be quite significant.). Calculating the TRR we find

(annual return)/(worst case) = 7% / 15% = 0.47

A house as an investment (Take 2)

Most people take out a loan to finance at least part of their real estate investments. Does borrowing part of the money change the analysis? Let us assume that we put $20,000 down on the house and borrow the remaining $80,000. Using the same assumptions as above, the Return on Equity is now 35% (7% of $100,000 or $7,000 divided by $20,000). The tail risk is a loss of $15,000 (15% of $100,000)  which is 75% of $20,000. The TRR is

(annual return)/(worst case) = 35% / 75% = 0.47

As expected, this is the same number we found before. In these examples we implicitly assumed a zero interest rate, but qualitatively nothing changes at non-zero interest rates. If we take out an $80,000 loan from the bank we have to pay interest on it. If we pay cash for the house, we lose the returns from $80,000 of investable assets. Either way we are paying the cost of money as interest to the bank or as missed investment returns.

Buying a Private Company

Another common illiquid investment is a stake in a private company. The valuation of such a company is generally very much in the eye of the beholder. The TRR offers a way such an investment can be compared to an investment in the S&P 500 or anything else. Let us assume that we buy the company at a P/E of 2. Assuming that the same earnings continue in the future, our expected return would be 50%. The risk in this case can be estimated rather precisely. The worst thing that can happen is that the company goes bankrupt and 100% of the investment is lost. This leads us to

(annual return)/(worst case) = 50% / 100% = 0.5

Conclusion

Traditional risk measures only work for liquid instruments and outside of the abnormally frequent market crashes. The Tail Risk Ratio attempts to supplement traditional risk measures with a less rigorous approach that acknowledges these shortcomings.

The examples above show that the TRR can be used for illiquid instruments without any problems other than the general difficulty of making good estimates of tail risk. We can now compare our primary residence to an investment in the S&P 500 on an equal footing.

The main drawback of the TRR is that there is no consistent framework for making tail risk estimates. It is more of an art than a science and requires a fair amount of historical research. While one can never be sure that the tail risk estimates are pessimistic enough, this approach does provide way to think about risk and reward that highlights the aspects of risk that are usually ignored because they are not mathematically tractable. The TRR should be a part of any investment decision. Combined with traditional risk measures and other considerations it will lead to more robust and rational investment decisions base on a more complete understanding of the risks involved. .

Martin Gremm, PhD
Pivot Point Advisors, LLC
(832) 778 7101

Math Refresher

In this article we will focus on volatility and the Sharpe Ratio as examples of traditional risk measures. Volatility is the standard deviation of returns, a measure for how much the value of the investment typically changes per time interval. The Sharpe Ratio is the return over a time interval divided by the volatility. A safer investment will have a higher Sharpe Ratio, i.e. return a higher multiple of a typical fluctuation. An investment with a higher Sharpe Ratio is said to have a higher risk-adjusted return.

In mathematical terms: